Nonuniqueness conjecture for quotient-neck singularities in Ricci flow

Let $k\geq3$. Consider Ricci flow on $S^1\times S^3/\mathbb{Z}_k$, and call a singularity a quotient-neck singularity when its tangent flow is $\mathbb{R}\times S^3/\mathbb{Z}_k$. Nonuniqueness conjecture. There exist a Ricci flow on $S^1\times S^3/\mathbb{Z}_k$ that forms a quotient-neck singularity with tangent flow $\mathbb{R}\times S^3/\mathbb{Z}_k$, and continuous nonuniquely modelled either on the Bryant soliton modulo $\mathbb{Z}_k$ or on Appleton's cohomogeneity-one soliton on the line bundle $O_{\mathbb{CP}^1}(-k)$. The source presents this as a proposed resolution of the remaining uniqueness-versus-nonuniqueness problem in critical dimension four; its status is therefore treated as open despite the parser's resolved flag, which refers to the surrounding conditional discussion.